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<p>With the above integrating factor, (<a href="" class="xref" data-knowl="./knowl/eq2_7.html" title="Equation 2.1.7">(2.1.7)</a>) becomes</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_7.html ./knowl/eq2_10.html ./knowl/eq2_11.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_12.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_11.html ./knowl/eq2_13.html">
\begin{equation}
\begin{aligned}
&amp;\frac{\textrm{d}}{\textrm{d} x} (u(x) y)=u(x) q(x) \rightarrow u(x) y=\int u(x) q(x) \textrm{d} x+C,\\
&amp;\rightarrow y=\frac{\int u(x) q(x) \textrm{d} x+C}{u(x)},
\end{aligned}\tag{2.1.11}
\end{equation}
</div>
<p class="continuation">with <span class="process-math">\(u(x)\)</span> given by (<a href="" class="xref" data-knowl="./knowl/eq2_10.html" title="Equation 2.1.10">(2.1.10)</a>). Equation (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a>) involves an arbitrary constant <span class="process-math">\(C\)</span> and includes every solution to (<a href="" class="xref" data-knowl="./knowl/eq2_6.html" title="Equation 2.1.6">(2.1.6)</a>) and it is called the <dfn class="terminology">general solution</dfn>. Geometrically, (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a>) represents a family of curves, called integral curves. On the other hand, if we impose that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_7.html ./knowl/eq2_10.html ./knowl/eq2_11.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_12.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_11.html ./knowl/eq2_13.html">
\begin{equation}
y(x_0)=y_0,\tag{2.1.12}
\end{equation}
</div>
<p class="continuation">which is called an initial condition. Then under (<a href="" class="xref" data-knowl="./knowl/eq2_12.html" title="Equation 2.1.12">(2.1.12)</a>), one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_7.html ./knowl/eq2_10.html ./knowl/eq2_11.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_12.html ./knowl/eq2_6.html ./knowl/eq2_11.html ./knowl/eq2_11.html ./knowl/eq2_13.html">
\begin{equation}
y_0=\frac{\int u(x) q(x) \textrm{d} x\Big|_{x=x_0}+C}{u(x_0)},\tag{2.1.13}
\end{equation}
</div>
<p class="continuation">which determines <span class="process-math">\(C\text{.}\)</span> <dfn class="terminology">ODE (<a href="" class="xref" data-knowl="./knowl/eq2_6.html" title="Equation 2.1.6">(2.1.6)</a>) together with initial condition (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a>) is called an initial value problem. The solution (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a>) with <span class="process-math">\(C\)</span> determined by (<a href="" class="xref" data-knowl="./knowl/eq2_13.html" title="Equation 2.1.13">(2.1.13)</a>) is called the particular solution.</dfn></p>
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